Problem: Of all eligible voters in your county, $70\%$ are currently registered to vote. A recent poll predicts that the relationship between $V$, the percentage of all eligible voters who are registered to vote, and $t$, the number of years from now will be modeled by the following equation. $V=100-30\cdot e^{-0.04t}$ In how many years will $80\%$ of all eligible voters in your county be registered to vote? Give an exact answer expressed as a natural logarithm. years
Explanation: Thinking about the problem We want to know how many years, $t$, it will take for the percent of registered voters, $V$, to reach $80\%$. So we need to find the value of $t$ for which $V=80$. Substituting $80$ in for $V$ in the model gives us the following equation. $80=100-30\cdot e^{-0.04t}$ Solving the equation We can solve the equation as shown below. $\begin{aligned}100-30\cdot e^{-0.04t}&=80\\\\ -30\cdot e^{-0.04t}&=-20\\\\ e^{-0.04t}&=\dfrac{2}{3}\\\\ -0.04t&=\ln\left(\dfrac{2}{3}\right)\\\\ t&=-25\cdot \ln\left(\dfrac{2}{3}\right)\\\\ \end{aligned}$ It will take $-25\cdot \ln\left(\dfrac{2}{3}\right)$ years for the percent of registered voters in your county to reach $80\%$. The expression above represents an exact solution to the equation. We can use a calculator to approximate the value of the expression, but this will be a rounded inexact answer. The answer The answer is $-25\cdot \ln\left(\dfrac{2}{3}\right)$ years.